CS 322: (Social and Information) Network Analysis Jure Leskovec Stanford University

Transcription

1 CS 322: (Social and Information) Network Analysis Jure Leskovec Stanford University

2 Probabilistic models of networkdiffusion How cascades spread in real life: Viral marketing Blogs Group membership Last 20 minutes: mid term course evaluation 2

3 How do viruses/rumors propagate? Will a flu like virus linger, or will it become extinct? (Virus) birth rate β: probability thanan infected neighbor attacks (Virus) death rate δ: probability that an infected node heals N 1 Prob. β Prob. δ N N 2 Healthy Infected N 3 10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis 3

4 General scheme for epidemic models: S susceptible E exposed I infected R recovered Z immune 4

5 Number of no odes Assuming perfect mixing, i.e., a network is a complete graph The model dynamics: time Susceptible Infected Recovered 5

6 Susceptible Infective Susceptible (SIS) model Cured nodes immediately become susceptible Virus strength : s = β / δ Infected by neighbor with prob. β Susceptible Cured internally with prob. δ Infective 10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis 6

7 Number of nodes Assuming perfect mixing (complete graph): ds N time dt di dt SI I SI I Susceptible Infected 7

8 Epidemic threshold of a graph is a value of t, such that: If strength s = β / δ < t epidemic can not happen (it eventually dies out) Given a graph compute its epidemic threshold 10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis 8

9 What should t depend on? avg. degree? and/or highest degree? and/or variance of degree? and/or third moment of degree? and/or diameter? 9

10 [Wang et al. 2003] We have no epidemic if: (Virus) Death rate Epidemic threshold β/δ < τ = 1/ λ 1,A (Virus) Birth rate largest eigenvalue of adj. matrix A λ 1,AA alone captures the property of the graph! 10

11 [Wang et al. 2003] N umber of Infected Nodes Oregon β = Time δ: ,900 nodes and 3, 31,180 edges β/δ > τ (above threshold) β/δ = τ (at the threshold) β/δ < τ (below threshold) 11

12 Does it matter how many people are initially infected? 12

13 [Leskovec et al., SDM 07] Bloggers write posts and refer (link) to other posts and the information propagates 13

14 Posts Blogs Information cascade Time ordered hyperlinks Dt Data Blogs: We crawled 45,000 blogs for 1 year 10 million posts and 350,000 cascades 14

15 [Leskovec et al., TWEB 07] Senders and followers of recommendations receive discounts on products 10% credit 10% off Data Incentivized Viral Marketing program 16 million recommendations 4 million people 500,000 products 15

16 [Backstrom et al., KDD 06] Use social networks where people belong to explicitly defined groups Each group defines a behavior that diffuses Data LiveJournal: On line blogging community with friendship links and user defined groups Over a million users update content each month Over 250,000 groups to join 16

17 Prob. of adoption depends on the number of friends who have adopted [Bass 69, Granovetter 78] What is the shape? Distinction has consequences for models and algorithms Prob. of adoptio on Prob. of adoptio on k = number of friends adopting Diminishing returns? k = number of friends adopting Critical mass? 17

18 [Leskovec et al., TWEB 07] Prob bability of purcha asing DVD recommendations (8.2 million observations) # recommendations received 18

19 [Backstrom et al., KDD 06] LiveJournal community membership Pr rob. of jo oining k (number of friends in the community) 19

20 [Kossinets Watts 06] Sending network of large university Prob. of a link as a function of # of common friends mail ob. of em Pro k (number of common friends) 20

21 For viral marketing: We see that node v receiving the i th recommendation and then purchased the product For communities: At time t we see the behavior of node v s friends Questions: When did v become aware of recommendations or fi friends d bh behavior? When did it translate into a decision by v to act? How long after this decision did v act? 21

22 Dependence on number of friends Consider: connectedness of friends x and y have three friends in the group x s friends are independent y s fi friends are all connected td x y Who is more likely to join? 22

23 Competingsociological theories x y Information argument [Granovetter 73] Social capital argument [Coleman 88] Information argument: Unconnected friends give independent support Social capital argument: Safety/truest advantage in having friends who know each other 23

24 [Backstrom et al. KDD 06] LiveJournal: 1 million users, 250,000 groups Social capital argument wins! Prob. of joining increases with adjacent members. 24

25 Large anonymous online retailer (June 2001 to May 2003) 15,646, recommendations dti 3,943,084 distinct customers 548, products recommended Products belonging to 4 product groups: books DVDs music VHS 25

26 t 1 < t 2 < < t n t 3 legend bought but didn t receive a discount t 1 bought and received a discount t 2 received a recommendation but didn t buy t 4 26

27 There are relatively few DVD titles, but DVDs account for ~ 50% of recommendations. Recommendations per person DVD: 10 books and music: 2 VHS: 1 Recommendations per purchase books: 69 DVDs: 108 music: 136 VHS: 203 Overall there are 3.69 recommendations per node on 3.85 different products. Music recommendations reached about the same number of people p as DVDs but used only 1/5 as many recommendations Book recommendations reached by far the most people 2.8 million. All networks have a very small number of unique edges. For books, videos and music the number of unique edges is smaller than the number of nodes the networks are highly disconnected 27

28 What role does the product category play? products customers recommenda- buy + get buy + no edges tions discount discount Book 103,161 2,863,977 5,741,611 2,097,809 65,344 17,769 DVD 19, ,285 8,180, ,341 17,232 58,189 Music 393, ,148 1,443, ,738 7,837 2,739 Video 26, , , , Full 542,719 3,943,084 15,646, ,153,676 91,322 79,164 high low people recommendations 28

29 Some products are easier to recommend than others product category number of buy bits forward recommendations Book 65,391 15, DVD 16,459 7, Music 7,843 1, Video Total 90,602 25, percent 29

30 Does sending more recommendations influence more purchases? 7 6 ber of Purcha ases Num Outgoing Recommendations 30

31 What is the effectiveness of subsequent recommendations? ying 0.06 Prob bability of bu Exchanged recommendations 31

32 consider successful recommendations in terms of av. # senders of recommendations per book category av. # of recommendations accepted books overall have a 3% success rate (2% with discount, 1% without) lower than average success rate (significant at p= level) fiction romance (1.78), horror (1.81) teen (1.94), children s books (2.06) comics (2.30), sci fi (2.34), mystery and thrillers (2.40) nonfiction sports (2.26) home & garden (2.26) travel (2.39) higher than average success rate (statistically significant) professional & technical medicine (5.68) professional & technical (4.54) engineering (4.10), science (3.90), computers & internet (3.61) law (3.66), business & investing (3.62) 32

33 47,000 customers responsible for the out of 16 million recommendations in the system 29% success rate per recommender of an anime DVD Giant component covers 19% of the nodes Overall, recommendations for DVDs are more likely to result in a purchase (7%), but the anime community stands out 33

34 Variable transformation Coefficient const *** # recommendations ln(r) *** # senders ln(n s ) *** # recipients ln(n r ) *** product price ln(p) *** # reviews ln(v) *** avg. rating ln(t) * R significance at the 0.01 (***), 0.05 (**) and 0.1 (*) levels 34

35 94% of users make first recommendation without having received one previously Size of giant connected component increases from 1% to 2.5% of the network (100,420 users) small! Some sub communities are better connected 24% out of 18,000 users for westerns on DVD 26% of 25,000 for classics on DVD 19% of 47,000 for anime (Japanese animated film) on DVD Others are just as disconnected 3% of 180,000 home and gardening 2 7% for children s and fitness DVDs 35

36 Products suited for Viral Marketing: small and tightly knit community few reviews, senders, and recipients but sending more recommendations helps pricey products rating doesn t play as much of a role Observations for future diffusion models: purchase decision more complex than threshold or simple infection influence saturates as the number of contacts expands links user effectiveness if they are overused Conditions for successful recommendations: professional and organizational contexts discounts on expensive items small, tightly knit communities 36

37 How big are cascades? What are the building blocks of cascades? Medical guide book 10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis DVD 37

38 Given a (social) network A process by spreading over the network creates a graph (a tree) Social network Cascade (propagation graph) Let s count cascades 38

39 is the most common cascade subgraph It accounts for ~75% cascades in books, CD and VHS, only 12% of DVD cascades is 6 (1.2 for DVD) times more frequent than For DVDs is more frequent than Chains ( ) are more frequent than is more frequent tthan a collision i ( ) (but collision has less edges) Late split ( ) is more frequent than 39

40 Stars ( no propagation ) Bipartite cores ( common friends ) Nodes having same friends 40

41 steep drop off books 10 6 = 1.8e6 x Co ount very few large cascades x = Cascade size (number of nodes) 41

42 DVD cascades can grow large Possibly as a result of websites where people sign up to exchange recommendations Count ~ x shallow drop off fat tail a number of large cascades x = Cascade size (number of nodes) 42

43 The probability of observing a cascade on x nodes follows: p(x) ~ x 2 Count x = Cascade size (number of nodes) 43

44 Cascade sizes follow a heavy tailed distribution Viral marketing: Books: steep drop off: power law exponent 5 DVDs: larger age cascades: exponent epoe 1.55 Blogs: Power law exponent 2 However, e it is not a simple peba branching gpocess process A simple branching process (a on k ary tree): Every node infects each of k of its neighbors with prob. p gives exponential cascade size distribution Questions: What role does the underlying social network play? Can make a step towards more realistic cascade generation (propagation) model? 44

45 1) Randomly pick blog to 2) Infect each in linked infect, add to cascade. neighbor with probability 1 B B 2 B 1 1 B B 3 B ) Add infected neighbors to cascade. 1 B 1 1 B B 3 B 4 B 1 B 1 B 1 B 4 B 3 B 4 4) Set node infected in (i) to uninfected. 1 B 1 1 B 2 B B 4 B1 B4 45

46 Generative model produces realistic cascades β=0.025 Count Cascade size Count Cascade node in degree Cou nt Co ount Most frequent cascades Size of star cascade Size of chain cascade 46

47 Blogs information epidemics Which are the influential/infectious blogs? Viral marketing Who arethe trendsetters? Influential people? Disease spreading Where to place monitoring stations to detect epidemics? 47